Name: Formal definition of limits Part 4: using the definition | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T14:15:28.000Z
Duration: 8 min 30 s
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In the last video, we took our first look

Adverbs of manner

at the epsilon-delta definition of limits, which essentially

says if you claim that the limit of f of x as x approaches C

is equal to L, then that must mean by the definition

that if you were given any positive epsilon that it

essentially tells us how close we want f of x to be to L.

greater than 0, which is essentially telling us

our distance from C such that if x is within delta of C,

then f of x is going to be within epsilon of L.

And what we will do in this video is use it

and to rigorously prove that a limit actually exists.

So, right over here, I've defined a function, f of x.

It's equal to 2x everywhere except for x equals 5.

So it's 2x everywhere for all the other values of x,

but when x is equal to 5, it's just equal to x.

At x is equal to 5, it's not along the line 2x.

And if I were to ask you what is the limit of f

And so, you might fairly intuitively make the claim

that the limit of f of x as x approaches 5 really

the epsilon-delta definition to actually prove it.

And the way that most of these proofs typically go

And then we essentially try to come up with a way

that given any epsilon, we can always come up with a delta.

to describe our delta as a function of epsilon,

that is defined for any positive epsilon.

put into our little formula or little function box.

And if I can do that for any epsilon that'll always give you

a delta, where this is true, that if x is within delta of C,

then the corresponding f of x is going to be within epsilon

So let's think about being within delta of our C.

So, let's think about this right over here is 5 plus delta,

So that's our range we're going to think about.

We're going to think about it in the abstract at first.

with a formula for delta in terms of epsilon.

So how could we describe all of the x's that are in this range

that are within delta of 5, but not necessarily equal to 5.

They're within a range of C, but not equal to C.

Well, that's going to be all of the x's that satisfy x minus 5

That describes all of these x's right over here.

And now, what we're going to do, and the way these proofs

to manipulate the left-hand side of the inequality,

so it starts to look something like this,

of the inequality is going to be expressed in terms of delta.

And then we can essentially say well, look.

of delta and the left-hand side looks just

like that, that really defines how we can express delta

If that doesn't make sense, bear with me.

So, if we want x minus 5 to look a lot more like this, when

x is not equal to 5-- in all of this, this whole interval,

x is not equal to 5-- f of x is equal to 2x,

So if we could somehow get this to be 2x minus 10,

to multiply both sides of this inequality by 2.

And 2 times the absolute value of something,

that's the same thing as the absolute value of 2 times

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Formal definition of limits Part 4: using the definition | AP Calculus AB | Khan Academy

literally

essentially

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